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We calculate the Laplace transform of the cut-and-join equation of Goulden,Jackson and Vakil. The result is a polynomial equation that has the topologicalstructure identical to the Mirzakhani recursion formula for the Weil-Peterssonvolume of the moduli space of bordered hyperbolic surfaces. We find that thedirect image of this Laplace transformed equation via the inverse of theLambert W-function is the topological recursion formula for Hurwitz numbersconjectured by Bouchard and Marino using topological string theory.

The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers